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# error function

The *error function* ${\rm erf}\colon\mathbb{C}\to\mathbb{C}$ is defined as follows:

${\rm erf}(z)={2\over\sqrt{\pi}}\int_{0}^{z}e^{{-t^{2}}}\,dt$ |

The *complementary error function* ${\rm erfc}\colon\mathbb{C}\to\mathbb{C}$ is defined as

${\rm erfc}(z)={2\over\sqrt{\pi}}\int_{z}^{\infty}e^{{-t^{2}}}\,dt$ |

The name “error function” comes from the role that these functions play in the theory of the normal random variable. It is also worth noting that the error function is a special case of the confluent hypergeometric functions and of the Mittag-Leffler function.

Note. By Cauchy integral theorem, the choice path of integration in the definition of ${\rm erf}$ is irrelevant since the integrand is an entire function. In the definition of ${\rm erfc}$, the path may be taken to be a half-line parallel to the positive real axis with endpoint $z$.

Defines:

complementary error function

Related:

AreaUnderGaussianCurve, ListOfImproperIntegrals, UsingConvolutionToFindLaplaceTransform

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

33B20*no label found*

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